Everkleen Pool Services (EPS) provides weekly swimming pool maintenance in Jeddah. Dozens of firms provide this service. The service is standardized; each company cleans the pool and maintains the proper levels of chemicals in the water. The service is typically sold as a four-month summer contract. The market price for the 4-month service contract is US$115.
EPS has fixed costs of US$3500. The manager has estimated the following marginal cost function for EPS, using data for the last two years:
MC = 125 – 0.42Q + 0.0021Q2
Where MC is measured in dollars and Q is the number of pools serviced each summer. Each of the estimated coefficients is statistically significant at the 95 percent confidence level.
1-Given the estimated marginal cost function, what is the average variable cost function for EPS?
2-At what output level does AVC reach its minimum value? What is the value of AVC at its minimum point?
3-Should the manager of EPS continue to operate, or should the firm shut down? Explain.
4-The manager of EPS finds two output levels that appear to be optimal. What are these levels of output and which one is actually optimal?
5-How much profit (or loss) can the manager of EPS expect to earn?
6-Suppose that EPS fixed costs rise to US$4000. How does this affect the optimal level of output? Explain.
1- To find the average variable cost function (AVC), we need to divide the total variable cost (TVC) by the output level (Q). The TVC is the sum of all the variable costs that change with the level of output. In this case, the variable cost is given by the marginal cost function (MC).
AVC = TVC / Q
To find TVC, we integrate the marginal cost function:
TVC = ∫(MC)dQ
Integrating MC = 125 - 0.42Q + 0.0021Q^2, we get:
TVC = 125Q - 0.42(Q^2)/2 + 0.0021(Q^3)/3
Simplifying this equation, we get:
TVC = 125Q - 0.21Q^2 + 0.0007Q^3
Now we can calculate the average variable cost function by dividing TVC by Q:
AVC = (125Q - 0.21Q^2 + 0.0007Q^3) / Q
Simplifying further:
AVC = 125 - 0.21Q + 0.0007Q^2
2- To find the output level at which AVC reaches its minimum value, we need to differentiate the AVC function with respect to Q and set the derivative equal to zero. Then solve for Q.
d(AVC)/dQ = -0.21 + 0.0014Q = 0
0.0014Q = 0.21
Q = 150
At an output level of 150, AVC reaches its minimum value.
To find the value of AVC at its minimum point, substitute Q=150 into the AVC equation:
AVC = 125 - 0.21(150) + 0.0007(150^2)
AVC = 125 - 31.5 + 0.0007(22500)
AVC = 93.5 + 15.75
AVC = 109.25
3- To determine whether EPS should continue operating or shut down, we need to compare the average variable cost (AVC) to the market price of the service contract.
If AVC is greater than the market price of US$115, it means that the cost of producing one unit of the service is higher than the revenue earned from selling it. In this case, the firm should shut down because it would be making a loss on each unit.
If AVC is less than the market price, it means that the revenue from selling the service is greater than the cost of producing it. In this case, the firm should continue operating and would be making a profit on each unit.
If AVC is equal to the market price, the firm would be breaking even, earning zero profit but covering all the costs.
4- The manager of EPS finds two output levels that appear to be optimal. To determine the truly optimal level of output, we need to compare the marginal cost (MC) to the market price.
If MC is less than the market price, it means that the firm can increase its profit by producing more units of the service. Therefore, the higher output level would be optimal.
If MC is greater than the market price, it means that the cost of producing an additional unit exceeds the revenue earned from selling it. In this case, the firm should reduce its output level to increase profitability. Therefore, the lower output level would be optimal.
By comparing the MC to the market price of US$115, we can determine which output level is truly optimal.
5- To calculate the profit or loss, we need to subtract the total cost (TC) from the total revenue (TR). TR is given by the market price multiplied by the output level (Q), while TC is the sum of fixed costs (FC) and total variable costs (TVC).
TR = Price * Q = US$115 * Q
TVC = 125Q - 0.21Q^2 + 0.0007Q^3 (Calculated previously)
FC = US$3500
TC = FC + TVC
Profit (or Loss) = TR - TC
6- If the fixed costs rise to US$4000, it would affect the optimal level of output. With higher fixed costs, the breakeven point would be higher. The firm would need to produce a higher output level to cover the increased fixed costs and achieve profitability.
The new optimal level of output would be determined by comparing the MC to the market price, as explained in question 4. With the higher fixed costs, the profitability threshold would be higher, affecting the decision on the optimal level of output.